Weak-Form Recovery of Stochastic Generators and Dynamical Invariants

Spectral gaps, Kramers escape rates, and position-dependent relaxation timescales are dynamical invariants encoded in the infinitesimal generator $\text{Lop}$ of a stochastic flow. We show that weak projection of the governing Itô SDE onto temporal test functions produces an endogeneity bias of order $O(T{\text{dt}}^{3\mathrm{/}2})$ that grows with the observation window and cannot be eliminated by additional data. Projecting instead onto spatial Gaussian kernels removes the bias exactly: ${\mathcal{F}}_{t_n}$-measurability and the tower property guarantee unbiased regression rows at every step. The resulting framework jointly identifies the drift $b(x)$ and diffusion $a(x)$ from a single sparse regression, producing an explicit symbolic enerator amenable to spectral analysis. Validation on three benchmark systems yields coefficient errors below 5%, stationary-density total-variation distances below 0.01, and autocorrelation functions that faithfully reproduce true relaxation timescales.